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Φ
be an EIM for the elements of which there exists an evaluating function
. Then
⎧
⎨
Φ(
a
)
k
1
,
l
ρ(
1
)
Φ(
a
)
k
2
,
l
ρ(
2
)
...Φ(
a
)
k
m
,
l
ρ(
m
)
,
[
ρ(
1
),...,ρ(
m
)
]
if
m
=
car d
(
K
)
≤
car d
(
L
)
=
n
per
Φ
(
A
)
=
,
⎩
Φ(
a
)
k
σ(
1
)
,
l
1
Φ(
a
)
k
σ(
2
)
,
l
2
...Φ(
a
)
k
σ(
n
)
,
l
n
,
[
σ(
1
),...,σ (
n
)
]
=
(
)
≥
(
)
=
if
m
car d
K
car d
L
n
where
ρ
:{
1
,
2
,...,
m
}→{
1
,
2
,...,
n
}
is a bijection from
{
1
,
2
,...,
m
}
in
{
1
,
2
,...,
n
}
and
σ
:{
1
,
2
,...,
n
}→{
1
,
2
,...,
m
}
is a bijection from
{
1
,
2
,...,
n
}
in
.
And yet, we can keep the concept of a determinant adding additional condition.
For brevity, here we discuss only the case of square EIM, i.e., the sets
K
and
L
have
equal number of elements. Let us fix the order of the elements of set
K
as a vector
[
{
1
,
2
,...,
m
}
k
1
,...,
k
m
]
. Each of its permutations
[
k
ρ(
1
)
,...,
k
ρ(
m
)
]
will be estimated as odd
or even about the vector
[
k
1
,...,
k
m
]
. Then, the determinant of the EIM
A
can be
defined by
det
Φ,
[
k
1
,...,
k
m
]
(
A
)
)
[
k
1
,...,
k
m
]
Φ(
=
(
−
1
a
)
k
1
,
l
ρ(
1
)
Φ(
a
)
k
2
,
l
ρ(
2
)
...Φ(
a
)
k
m
,
l
ρ(
m
)
.
[
ρ(
1
),...,ρ(
m
)
]
3.7 Transposed EIM
Let the EIM
A
be given as above. Then its Transposed EIM has the form
k
1
...
k
i
...
k
m
l
1
a
l
1
,
k
1
...
a
l
1
,
k
i
...
a
l
1
,
k
m
.
.
.
.
.
. . .
...
A
=[
L
,
K
,
{
a
l
j
,
k
i
}] =
.
l
j
a
l
j
,
k
1
...
a
l
j
,
k
i
...
a
l
j
,
k
n
.
.
.
.
.
. . .
...
l
n
a
l
n
,
k
1
...
a
l
n
,
k
i
...
a
l
n
,
k
m
The geometrical interpretation is
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